Graded Differential Geometry of Graded Matrix Algebras

TL;DR

This paper develops a comprehensive graded differential geometry framework for matrix algebras with a Z2 grading, establishing their structure as noncommutative graded manifolds with universal calculus properties.

Contribution

It introduces and investigates graded Cartan calculus, symplectic structures, and connections on graded matrix algebras, proving their universality and manifold-like properties.

Findings

Proves the universality of the graded derivation-based first-order calculus.

Shows ${f M}(n|m)$ behaves as a noncommutative graded manifold.

Establishes the cohomology of ${f M}(n|m)$ matches its body.

Abstract

We study the graded derivation-based noncommutative differential geometry of the $Z_2$-graded algebra ${\bf M}(n| m)$ of complex $(n+m)\times(n+m)$-matrices with the ``usual block matrix grading'' (for $n\neq m$). Beside the (infinite-dimensional) algebra of graded forms the graded Cartan calculus, graded symplectic structure, graded vector bundles, graded connections and curvature are introduced and investigated. In particular we prove the universality of the graded derivation-based first-order differential calculus and show, that ${\bf M}(n|m)$ is a ``noncommutative graded manifold'' in a stricter sense: There is a natural body map and the cohomologies of ${\bf M}(n|m)$ and its body coincide (as in the case of ordinary graded manifolds).